Biomedical Engineering Reference
In-Depth Information
FIGURE 4-49
Classic configuration
of a PID controller.
In terms of motor control, the proportional plus integral controller returns the system
to stability, but it is still highly underdamped.
4.8.4 Proportional-Integral-Derivative Controller
Proportional-integral-derivative (PID) controllers are also known as three-terminal con-
trollers and have the transfer function
K
i
s
+
K
d
s
G
c
(
s
)
=
K
p
+
(4.94)
The system block diagram is shown in Figure 4-49.
As with the proportional plus integral (PI) controller, equation (4.94) can be rewritten
in terms of the integral time constant,
τ
i
=
K
p
/
K
i
, and a derivative time constant,
τ
d
=
K
d
/
K
p
,
K
p
1
τ
i
s
+
τ
d
s
1
G
c
(
s
)
=
+
(4.95)
In this case the forward transfer function is
+
τ
i
τ
p
s
2
K
p
(τ
i
s
+
1
)
G
o
(
s
)
=
G
p
(
s
)
(4.96)
τ
i
s
It can be seen that this has increased the number of zeros by two and the number of poles
by one as well as increasing the overall system order by one.
In the motor control example, a PID controller block replaces the proportional gain
block, as shown in Figure 4-50.
In this case the proportional gain,
K
p
1, and the
derivative gain,
K
d
, are set to a range of values. The step response for the PID controlled
position controller is shown in Figure 4-51.
It can be seen that the response is very similar to that for a proportional controller
when
K
d
=
10, the integral gain,
K
i
=
0. However, as the derivative gain increases, damping is increased and the
ringing decreases. The critically damped response occurs for
K
d
=
=
0
.
75, and above that
the response becomes overdamped.
